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Question

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The y-axis is labeled amount left parenthesis milligrams right parenthesis. The x-axis is labeled time left parenthesis hours right parenthesis.

Maria took 500 mg of medicine for her headache. The drug depletes in the blood stream at a rate of 20% per hour. The situation can be modeled by the exponential equation a(t)=500(0.8)t
, where a is the amount of medicine in the blood stream after t hours. Which one of the following statements is true about the graph of the function?

Option #1: As t increases, a increases slowly at first and then quickly.

Option #2: The y-intercept of the function is (0,500)
.

Option #3: The range includes all real numbers.

(1 point)

Answer 1

To evaluate the truth of the statements given about the graph of the function \( a(t) = 500(0.8)^t \), let's analyze each option:

Option #1: "As t increases, a increases slowly at first and then quickly."

• This statement is not true. The function \( a(t) = 500(0.8)^t \) is a decreasing exponential function. As time \( t \) increases, the value of \( a(t) \) decreases, not increases. Therefore, this option is false.

Option #2: "The y-intercept of the function is (0,500)."

• To find the y-intercept, we evaluate \( a(0) \): \[ a(0) = 500(0.8)^0 = 500(1) = 500 \] Thus, the y-intercept is indeed \( (0, 500) \). Therefore, this option is true.

Option #3: "The range includes all real numbers."

• The range of \( a(t) = 500(0.8)^t \) is \( (0, 500] \). As \( t \) increases, \( a(t) \) approaches 0 but never reaches negative values. Hence, this option is false.

Based on this analysis, the only true statement about the graph of the function is Option #2: "The y-intercept of the function is (0,500)."